Equivalence of flat connections and Fay identities on arbitrary Riemann surfaces

A flat connection on a Riemann surface with values in an infinite dimensional Lie algebra provides a systematic and effective tool for generating an infinite family of polylogarithms via iterated integrals. The recent literature offers different types of connections, in one or several variables, on compact Riemann surfaces with or without punctures, and in the meromorphic or single-valued categories. In this work, we show that the flatness conditions for the single-valued and modular DHS connection in multiple variables, which was introduced in the companion paper arXiv:2602.01461, are equivalent to the union of all the interchange and Fay identities among DHS integration kernels that were proven in arXiv:2407.11476. Based on the same combinatorial techniques, the flatness conditions on the multivariable Enriquez connection is shown to imply the union of all the interchange and Fay identities for Enriquez kernels.